Analysis of antennas in the presence of large composite 3-D structures with Linear Embedding via Green's Operators (LEGO) and a modified EFIE

Analysis of antennas in the presence of large composite 3-D

structures with Linear Embedding via Green's Operators

(LEGO) and a modified EFIE

Citation for published version (APA):

Lancellotti, V., Hon, de, B. P., & Tijhuis, A. G. (2010). Analysis of antennas in the presence of large composite 3-D structures with Linear Embedding via Green's Operators (LEGO) and a modified EFIE. In Proceedings of the Fourth European Conference on Antennas and Propagation (EuCap), 12-16 April 2010, Barcelona, Spain (pp. 1-5). Institute of Electrical and Electronics Engineers.

Document status and date: Published: 01/01/2010

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Analysis of Antennas in the Presence of Large

Composite 3-D Structures with Linear Embedding

via Green’s Operators (LEGO) and a Modified EFIE

V. Lancellotti, B. P. de Hon, A. G. Tijhuis

Department of Electrical Engineering, Technical University of Eindhoven P. O. Box 513, 5600 MB Eindhoven, The Netherlands

{v.lancellotti, b.p.d.hon, a.g.tijhuis}@tue.nl Abstract—We combine the linear embedding via Green’s

operators method with an electric field integral equation (EFIE) to solve the problem of an antenna system which radiates in close proximity of a large 3-D structure. Upon rearranging the relevant equations we include the contribution of the large structure into the EFIE posed over the antenna surface — which results in a “modified” EFIE. The latter can be solved by MoM and direct methods as long as the antenna system is not too large. Therefore, the present approach is superior to the bare MoM, which (for the same problem) would yield a huge matrix to be inverted through iterative methods. We provide validation of the proposed strategy as well as an example of application to a real-life antenna operating nearby a finite frequency selective surface.

I. BACKGROUND AND OVERVIEW

Thanks to the availability of modern powerful and fast computers, nowadays the solution of large (say, ten or more wavelengths in diameter) scattering and radiation problems has become a common task for which indeed many numerical methods have been devised over the past decades. A widely popular approach, based on integral equations, is the Multi-Level Fast Multipole Algorithm (MLFMA) [1], [2]. Used to-gether with the Method of Moments (MoM) [3], the MLFMA allows one to fill the relevant MoM matrix fast and to account efficiently for both large and small scale details of a given object. When applied to large, complex and arbitrarily shaped structures, though, the MLFMA may lead to algebraic systems with millions of unknowns and full matrices, and hence one has to resort to iterative solvers. What’s more, in cases where the structure of interest — albeit large — is mostly comprised of many identical objects arranged in a regular repetitive pattern, the flexibility of the MLFMA is quite an overkill.

For such problems, a far better strategy is one that can take advantage of the inherent “discrete” nature of the structure as well as of its (finite) translational symmetry, if any. Methods following this line of thought are: The Synthetic Function eXpansion (SFX) [4], [5], the Characteristic Basis Function Method (CBFM) [6], [7], the Equivalence Principle Algorithm (EPA) [8] along with its improved version the Tangential-Equivalence Principle Algorithm (T-EPA) [9], and the Linear Embedding via Green’s Operators (LEGO). The latter was first proposed for 2-D electromagnetic band-gap (EBG) devices in [10] and recently fully extended to 3-D problems [11]–[15] by these authors.

+

∂

D

ND OBJECTS AND LEGO BRICKS k

D

∂

D

Fig. 1. Sketch of the antenna problem and LEGO approach: a multiport antenna system (the caseNP = 2 is shown) operates in the presence of a large

3-D composite structure comprised ofNDmetallic or dielectric objects. Each object is enclosed within a LEGO brickD_k,k = 1, . . . , ND, characterized by its scattering operatorS_kk.

Although notable differences exist among the numerical techniques listed above, nevertheless they may be termed domain decomposition methods (DDMs). In fact they all share the basic idea of dividing a large 3-D structure into a number of (possibly interconnected) sub-domains, on which sets of locally entire-domain functions are introduced to expand the unknown. Since normally the total number of functions so defined happens to be far smaller than the original number of sub-domain functions required by the MoM, the corresponding algebraic system matrix can be inverted using direct solvers.

In this paper we address the problem of a perfect electric conducting (PEC) multiport antenna system that is to radiate in the presence of a large 3-D composite structure. The latter, sketched in Fig. 1 along with a two-port antenna, may be a frequency selective surface (FSS), an EBG structure or a high impedance surface, to name but a few practical applications. As anticipated, the electromagnetic (EM) problem in Fig. 1 is more efficiently treated by means of a DDM, such as LEGO, rather than the MLFMA. In LEGO [12] we tackle the EM

scattering by an aggregate of ND bodies (immersed in a

ho-mogeneous host medium) by enclosing each object within an arbitrarily-shaped bounded domainD_k(brick),k = 1, . . . , ND

(see Fig. 1). Then, by invoking Love’s Equivalence Principle [16], we characterize the bricks electromagnetically by means

of scattering operators S_kk, which we subsequently combine

the extension to be discussed, we also need to pose an electric

field integral equation (EFIE) [3] on the antenna surface SA

to be solved for the current densityJA.

Unlike the scattering problems we considered in previous papers [12], [15] (where the incident fields were radiated either by sources at infinity or by elemental dipoles) in Fig. 1 a real antenna acts as a source for the structure. However, since we are interested in structures comparatively large with respect to the antenna, we formulate the problem so as to incorporate the effect of the structure into the EFIE. For this reason we call the resulting equation a “modified” EFIE. A key step in our approach is the usage of LEGO and the eigencurrent expansion method (EEM) [12], [15] for determining the contribution of the structure. The overall procedure turns out convenient chiefly because, on the one hand, the contribution of the large structure (the calculation of which may be time consuming) has to be computed only once, whereas, on the other hand, the modified EFIE can be solved with the MoM and LU factorization, provided the antenna system is not too large.

The rest of the paper is organized as follows. The modified EFIE and its numerical solution are described in Section II and III, respectively. In Section IV we first discuss the validation of the numerical code we developed and then we use it to analyze a two-port antenna system radiating in the presence

of a finite FSS. An exp(jωt) time variation is assumed for

fields and sources, and it is implied throughout.

II. FORMULATION WITHLEGOAND A MODIFIEDEFIE

To tackle the EM problem in Fig. 1, we begin by applying LEGO [12] to the large composite structure. As a result, we end up with an integral equation involving the total inverse scattering operatorS−1, viz.,

S−1_qs_{= q}i

, (qs,i)k = qks,i, qks,i=

 Js,i k√η −Ms,i k/√η  , (1)

where η = μ/ε is the intrinsic impedance of the host

medium. The column vectors qs,i_k represent (equivalent) scat-tered and incident current densities on either side of∂Dk.

Secondly, on forcing the total tangential electric field to vanish over the antenna surface SA, we arrive at the EFIE

 Eg_{+ E}s₊ ND  k=1 Es k  tan = 0, onSA, (2)

whereEg is the known incident field due to delta-gap sources [17] existing at each antenna port, E_ks is the scattered field radiated byqs

kon∂D−k, andEsis the scattered field produced

by the currentJA onSA, viz. [Es_]

tan= ηLAJA, (3)

with LA the usual EFIE operator [3] normalized to η. To

specify the EM problem fully we have to supplement (1)-(3) with the coupling relations

qki =√η(Pkki )−1PkAJA, [E_ks]tan=√ηPAkqsk, (4)

which express the mutual interaction between the antenna and the large structure in Fig. 1. We refer to the operatorsP_kki ,PkA

andPAk as propagators, because in general they link current densities on a surface to tangential fields on another surface. For instance, Pi

kk (explicitly given in [12, Table I]) relateqik

to the incident tangential fields on ∂D+_k. Similarly, P_kA and PAk represent propagators fromSAto∂Dk and vice-versa. In particular, P_kA (PAk) is a2 × 1 (1 × 2) abstract matrix whose elements are integro-differential operators involving the dyadic Green’s function of the background medium [3].

Now, by plugging (3), (4) into (1), (2) and eliminating qs we obtain the following modified EFIE

η (LA+ LS) JA= −[Eg]tan, (5)

where the operator LS rigorously captures the effect of the

fixed large structure. By virtue of (1), (4) we can express LS formally as

LS= ΘASSTSA, (6)

where ΘAS and TSA are a row and a column vector,

respec-tively, with entries

(ΘAS)k = PAk, (TSA)k= (Pkki )−1PkA. (7)

Notice that to deriveS−1 in (1) andTSA in (7), we have to

compute S_kk andPi

kk only once if the bricks that model the

large structure are all identical to one another.

III. SOLUTION THROUGHMOMANDEEM

We solve (5) via standard MoM (in Galerkin’s form) com-bined with the EEM. To this end, we model the surfaces of the

ND bodies and the bricks and the antenna by means of 3-D

triangular-facet meshes with which we associate suitable sets of Rao-Wilton-Glisson (RWG) [18] vector elements.

Specifi-cally, we introduceNO (2NF) RWG functions on a body’s (a

brick’s) surface for representing qo (q_ks,i) as in [12], plus NA functions onSA for expandingJA, viz.

JA=

NA



l=1

hlIl. (8)

Then, by using (6), plugging (8) into (5) and testing it against {hl}Nl=1A through a symmetric inner product we obtain

η ([LA] + [LS]) [IA] = − [Eg] , (9)

[LS] = [ΘAS] [S] [TSA] , (10)

where with transparent notation each matrix represents the algebraic counterpart of the corresponding operator, and [IA] is a column vector storingIl. Finally,[Eg] is a column vector

with entries ([Eg_])

l=_Sld2r hl· Eg, where d2r denotes the

area element and the integration is carried out on the pair of adjacent triangles which constitute the support ofh_l. Once (9) has been solved, we go on to compute the scattered currents coefficients from

[qs_{] =}√

η [S] [TSA] [IA] , (11) which ensues from (1) and the first of (4).

-0.5 0 0.5 1 1.5 -0.5 0 0.5 -0.5 0 0.5 x [m] y [m] z [ m ] (a) -0.5 0 0.5 1 1.5 -0.5 0 0.5 -0.5 0 0.5 x [m] y [m] z [ m ] (b)

Fig. 2. For LEGO validation: (a) two PEC half-wavelength dipoles radiate in the presence of two PEC spheres; (b) to apply LEGO/EEM the spheres are enclosed inND= 2 cubic bricks. Antennas and spheres and bricks are

modelled by means of triangular facets on which RWG functions are defined.

From (10), (11) we see that the calculation of[LS] and [qs] entails the inversion of[S]−1, i.e., the algebraic counterpart of S−1_{. As pointed out in [12],[S]}−1_{(of size2N}

FND×2NFND)

may be relatively large whenND 1. Therefore we employ

the EEM to reduce [S]−1 to a far smaller matrix which we

can invert via LU factorization: We refer the reader to [12, Section IV] for the details. For the sake of clarity we just repeat the result, namely, the expression of[S]−1 in the basis of the eigencurrents [ ˆS]−1=  [ ˜SCC]−1[ ˜SCU]−1 [ ˜SUC]−1[ ˜SUU]−1  ≈  [ ˜SCC]−1 [0] [0] [ΛUU]−1  , (12) where the subscript C (U) stands for coupled (uncoupled). OnlyNCNDeigencurrents out of the grand total2NFND

hap-pen to be coupled and take part in the multiple scattering that occurs amongst the bricks. This contribution is captured by the reduced inverse scattering operator [ ˜SCC]−1 of rank NCND.

The remaining eigencurrents are uncoupled, as they do not interact with one another nor with any coupled eigencurrent, and hence their contribution amounts to the diagonal matrix [ΛUU]−1. The latter contains the reciprocal of the eigenvalues associated with the uncoupled eigencurrents and it is never computed as such [12], [14], [15].

Now, if we express[ΘAS], [TSA] as well in the basis of the eigencurrents and we employ (12), we can write (10) as

[LS] ≈  [ ˜ΘAC] [ ˜ΘAU] [ ˜SCC] [0] [0] [ΛUU]   [ ˜TCA] [ ˜TUA]  , (13)

which we can further reduce to

[LS] ≈ [ ˜ΘAC][ ˜SCC][ ˜TCA], (14) upon recalling that the eigenvalues associated with the un-coupled eigencurrents decay to zero [14]. In addition, from a physical standpoint, (14) states that the fields radiated by the uncoupled eigencurrents do not reach the antenna (nor are they reflected back from it), inasmuch as such fields are tightly confined around a brick’s boundary. Incidentally, an order reduction similar to (14) also applies to (11).

Notice that (14) is efficient because in general[ ˜SCC]−1is far

smaller than[S]−1 — which allows us to deal with relatively

large structures. More importantly, the analysis of different antenna systems in the presence of the same structure is greatly

0.4 0.5 0.6 −30 −25 −20 −15 −10 −5 0 L/λ0 Magnitude [dB] S₁₁ S₂₁ 0.4 0.5 0.6 0.7 −3 −2 −1 0 1 2 3 L/λ0 Phase [rad] S₁₁ S₂₁

Fig. 3. LEGO validation: scattering parameters (magnitude and phase) of the dipoles and the spheres in Fig. 2. The results of the LEGO/EEM approach

(•) are compared to the baseline MoM solution (−/− −).

facilitated in that to write (9) we only have to fill[LA], [ ˜ΘAC], [ ˜TCA]. Finally, for antenna systems of small to moderate size,

the number of functions NA required to expand JA is likely

not too large: As a result (9) can be inverted with direct methods.

IV. NUMERICAL RESULTS

We have further extended our code to implement (9). In order to validate the results, we have considered, among others, the simple antenna problem shown in Fig. 2(a). The multi-port

antenna consists of NP = 2 half-wavelength dipoles (length

L = 1 m, width w, w/L = 0.1) which irradiate two nearby

PEC spheres (radiusa, a/L = 0.25). We energize the system

by delta-gap sources located at the center of each dipole. For this problem we obtain a reference solution through a standard EFIE which we solve for the electric current density, say J, by the MoM with 717 RWG functions. Secondly, we

apply LEGO/EEM by embedding the spheres within ND= 2

cubic bricks, as shown in Fig. 2(b). The number of RWG

functions on each sphere is NO = 294, whilst 2NF = 1800;

the rank of[LA] + [LS] in (9) is NA= 30. We set the number

of coupled eigencurrents toNC= 30, thereby the size of the

reduced inverse scattering operator [ ˜SCC]−1 isNDNC = 60.

As we force the input voltages and (from J, JA) we

compute the currents flowing in each antenna port, the natural set of network parameters we can derive is the matrix of

short-circuit admittances, YA. From YA we pass to the scattering

matrix SA [19] with reference impedances ZR = 50 Ω at

each port. In Fig. 3 we have plotted (as a function of the electric lengthL/λ0) the reflection (S11) and the transmission

(S21) coefficients computed through LEGO/EEM (•) and the

standard EFIE (−/− −). The comparisons turn out excellent,

thus confirming the validity of the approach based on the modified EFIE (9) combined with (14).

As an example of application to real-life structures, we have probed a finite FSS [20] by means of two folded trapezoidal toothed log-periodic antennas (LPAs) [17]. Both the FSS and the LPAs (shown in Fig. 4) are immersed in free-space. The FSS is made of 200 infinitely thin PEC cross-dipoles (length L = 0.015 m, width w, w = L/6). The latter are distributed

−0.1 −0.05 0 0.05 0.1 0 0.05 0.1 0.15 0 0.05 0.1 0.15 x [m] y [m] z [m]

Fig. 4. Application example: a finite FSS (comprised of two layers of cross-dipoles) is probed by means of two folded log-periodic antennas.

-5 0 5 x 10-3 -0.01 0 0.01 -0.01 -0.005 0 0.005 0.01 x [m] y [m] z [ m ] (a) -5 0 5 x 10-3 -0.01 0 0.01 -0.01 -0.005 0 0.005 0.01 x [m] y [m] z [ m ] (b)

Fig. 5. For defining the FSS in Fig. 4: close-up of (a) a pair of cross-dipoles forming the unit cell and (b) the enclosing LEGO brick.

are arranged in a regular 10-by-10 rectangular lattice. Each LPA consists of 12 dipoles arranged in two arms which are tilted to form an angle of 40 degrees. To apply LEGO/EEM we pair adjacent cross-dipoles that lie on different planes [Fig. 5(a)] and we embed them inND= 100 bricks [Fig. 5(b)].

In this numerical experiment, NO = 216, 2NF = 1344,

NC = 25, NA = 1220, so that the sizes of the total and

the reduced inverse scattering operators are134400 × 134400

and2500×2500, respectively, whereas the rank of [LA]+[LS] is 1220.

We have computedSAfor the LPAs in the presence of the

FSS for 26 frequency samples evenly distributed from 7.5 to 10

GHz and for the separationd/L ∈ {40/3, 8, 20/3} between

the antennas. For reference we have also obtained SA in the

case when the FSS is removed. The entries of SA, namely,

S11 andS21 are compared in Fig. 6.

Apparently, the LPAs — which were designed to be well matched in the frequency band of interest — are not notably

affected by the FSS in the range L/λ0 ∈ [0.375, 0.425],

re-gardless of their reciprocal distance. This suggests that for the very same frequencies the FSS is practically transparent. To some extent, this behavior may be justified upon considering

0.4 0.45 0.5 -30 -20 -10 0 |S11 | [ dB ] 0.4 0.45 0.5 -40 -30 -20 -10 |S21 | [ dB ] d/L = 40/3 0.4 0.45 0.5 -30 -20 -10 0 |S11 | [ dB ] 0.4 0.45 0.5 -30 -20 -10 0 |S21 | [ dB ] d/L = 8 0.4 0.45 0.5 -30 -20 -10 0 L/λ₀ |S11 | [ dB ] 0.4 0.45 0.5 -30 -20 -10 0 L/λ₀ |S21 | [ dB ] d/L = 20/3

d

L

Fig. 6. Scattering parameters (|S11|, |S21|) of the system in Fig. 4 versus

the cross-dipole electric lengthL/λ0 and for different separationsd of the

LPAs: (−) LPAs without the FSS, (− −) LPAs in the presence of the FSS,

(· · · ) 10% reflected power threshold. Inset: sketch of the FSS and the LPAs.

the transmitted and back-scattered radar cross section (RCS) of the FSS shown in Fig. 7. In fact, it is seen that the FSS

exhibits a minimum of reflection for L/λ0≈ 0.395.

Conversely, in the upper part of the frequency band, the degree of matching strongly depends on how far the LPAs are located away from the FSS. Contrary to expectation, though, a better matching does not result in an increase in transmitted power, and especially so for the most distant configuration (d/L = 40/3). What’s more, a better matching (which means that less power gets reflected back at the LPAs ports) seems at odds with the larger back-scattered RCS of the FSS in the same band (see Fig. 7). This observation, however, may not completely fit to the case at hand, for the LPAs are not in the far field region of the FSS. Therefore, as an explanation to

the conspicuous drop in the combined level of S11 andS21,

we speculate that in the FSS guided waves are excited that carry away the power radiated by the LPAs. To support our hypothesis, in Fig. 8 we have plotted sections of the radiation

pattern of the combined FSS-LPAs system for d/L = 40/3

at L/λ0 = 0.475. As expected, for φ ∈ {π, 3π/2} (i.e., in

the plane of the FSS) substantive radiation takes place in the z-direction.

0.38 0.4 0.42 0.44 0.46 0.48 0.5 25 30 35 40 45 50 55 L/λ₀ σ/L 2 [d B ] Back-scattered Transmitted _L s Ei k

Fig. 7. Normalized radar cross section of the FSS in Fig. 4 versus the cross-dipole electric lengthL/λ0. Inset: sketch of the FSS and geometrical quantities and incident plane wave.

frequency for the FSS-LPAs problem required less than 41 minutes on a Linux-based x86 64 workstation endowed with an Intel Xeon 2.66-GHz processor and 8-GB RAM. Of the

total computational time only ≈ 8 minutes were used to fill

[ ˜SCC]−1, thanks to the advantage achieved by exploiting the finite translational symmetry of the FSS, as in [12], [15]. Instead, a sizable fraction of the time was spent in filling[ ˜ΘAC] and[ ˜TCA], because we had to handle the interaction of each

brick (recall ND = 100) with the antennas and vice-versa

separately.

V. CONCLUSION

We have discussed a strategy (based on LEGO and a modified EFIE) to deal with antennas radiating in the presence of large 3-D composite structures. The key step is to include the effect of the structure into the EFIE on the antenna surface. From a numerical viewpoint the procedure stands out as a valid alternative to the bare MoM for both flexibility and time requirements. In fact, our approach applies to composite structures comprised of PEC and penetrable bodies as well, while our modified EFIE can be solved by MoM and direct methods. Further extension to structures comprising PEC bodies immersed in a finite dielectric medium is well under way.

ACKNOWLEDGMENTS

This research was supported by the post-doc fund un-der TU/e project no. 36/363450, and is performed in the framework of the MEMPHIS project (http://www.smartmix-memphis.nl/).

REFERENCES

[1] J. M. Song, C. C. Lu, and W. C. Chew, “MLFMA for electromagnetic scattering from large complex objects,” IEEE Trans. Antennas Propag., vol. 45, pp. 1488–1493, Oct. 1997.

[2] K. C. Donepudi, J. M. Song, J. M. Jin, G. Kang, and W. C. Chew, “A novel implementation of multilevel fast multipole algorithm for higher order Galerkin’s method,” IEEE Trans. Antennas Propag., vol. 48, pp. 1192–1197, Aug. 2000.

[3] A. F. Peterson, S. L. Ray, and R. Mittra, Computational methods for

electromagnetics. Piscataway: IEEE Press, 1998.

0.4 0.6 0.8 1 30 150 60 120 90 90 120 60 150 30 180 0 _{φ/π = 0} φ/π = 0.5 φ/π = 1 φ/π = 1.5 d/L = 40/3 L/λ₀ = 0.475

Fig. 8. Radiation pattern (natural units) of the LPAs in Fig. 4 in the presence of the FSS forL/λ0= 0.475 and d/L = 40/3.

[4] L. Matekovitz, V. A. Laza, and G. Vecchi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas

Propag., vol. 55, no. 9, pp. 2509–2521, Sep. 2007.

[5] H.-W. Yuan, S.-X. Gong, Y. Guan, and D.-Y. Su, “Scattering analysis of the large array antennas using the synthetic basis functions method,”

J. Electromagn. Waves and Appl., vol. 23, no. 2-3, pp. 309–320, 2009.

[6] R. Mittra and K. Du, “Characteristic basis function method for iteration-free solution of large method of moments problems,” Progress In

Electromagnetics Research B, vol. 6, pp. 307–336, 2008.

[7] J. Laviada, F. Las-Heras, M. R. Pino, and R. Mittra, “Solution of elec-trically large problems with multilevel characteristic basis functions,”

IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3189–3198, Oct.

2009.

[8] M. K. Li and W. C. Chew, “Wave-field interaction with complex structures using equivalence principle algorithm,” IEEE Trans. Antennas

Propag., vol. 55, no. 1, pp. 130–138, Jan. 2007.

[9] P. Yl¨a-Oijala and M. Taskinen, “Electromagnetic scattering by large and complex structures with surface equivalence principle algorithm,” Waves

in Random and Complex Media, vol. 19, no. 1, pp. 105–125, Feb. 2009.

[10] A. M. van de Water, B. P. de Hon, M. C. van Beurden, A. G. Tijhuis, and P. de Maagt, “Linear embedding via Green’s operators: A modeling technique for finite electromagnetic band-gap structures,” Phys. Rev. E, vol. 72, no. 5, pp. 1–11, Nov. 2005.

[11] V. Lancellotti, B. P. de Hon, and A. G. Tijhuis, “Electromagnetic mod-elling of large complex 3-D structures with LEGO and the eigencurrent expansion method,” in AP/URSI Int. Symp., Charleston, SC, June 2009. [12] ——, “An eigencurrent approach to the analysis of electrically large 3-D structures using linear embedding via Green’s operators,” IEEE Trans.

Antennas Propag., vol. 57, no. 11, pp. 3575–3585, Nov. 2009.

[13] ——, “A total inverse scattering operator formulation for the analysis of large 3-D structures,” in 11th ICEAA, Torino, ITALY, September 2009. [14] ——, “On the convergence of the eigencurrent expansion method applied to linear embedding via Green’s operators,” submitted November 2009, under review.

[15] ——, “Sensitivity analysis of 3-D composite structures through linear embedding via Green’s operators,” Progress In Electromagnetics

Re-search, vol. 100, pp. 309–325, Jan. 2010.

[16] I. Lindell, “Huygens’ principle in electromagnetics,” Science,

Measure-ment and Technology, IEE Proceedings, vol. 143, no. 2, pp. 103–105,

Mar. 1996.

[17] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: John Wiley & Sons, Inc., 1997.

[18] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982.

[19] D. Pozar, Microwave Engineering. New York: Wiley, 1998. [20] R. Mittra and V. V. S. Prakash, “Analysis of large finite frequency

selective surfaces embedded in dielectric layers,” in AP/URSI Int. Symp., vol. 1, San Antonio, TX, 2002, pp. 572–575.